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The state-norm estimation problem for the switched nonlinear system under a more generalized type of switching signals is investigated in this paper. The component subsystems consist of input/output-to-state stability (IOSS) and non-IOSS continuous systems simultaneously, whose activation time is lower and upper bounded by mode-dependent time-varying functions, respectively. The considered switching signals satisfy mode-dependent time-varying constraints, whose switching number can grow faster than those with constant average dwell time condition. Then based on multiple IOSS Lyapunov functions method, the existence condition and design method for the switched state-norm estimator are proposed. The obtained results present extensions compared with the existing results, which are applicable for the more general situations. Finally, the effectiveness and improvement of the proposed approach are illustrated by a numerical example. [ABSTRACT FROM AUTHOR]